Arm I: Patients receive dexrazoxane IV over 10-20 minutes, doxorubicin IV over 5-10 minutes, and cyclophosphamide IV over 30 minutes on days 1, 22, 43, and 64. Patients receive paclitaxel IV over 1 hour and trastuzumab (Herceptin) IV over 30-90 minutes on days 85, 92, 99, 106, 113, 120, 127, 134, 141, 148, 155, and 162. Approximately 1-2 weeks after completion of neoadjuvant chemotherapy, patients undergo breast conservation surgery, modified radical mastectomy, or mastectomy. Patients with unacceptable toxicity or locoregional disease progression may undergo surgery prior to week 24 (i.e., completion of neoadjuvant chemotherapy). Beginning 2-4 weeks after breast conservation surgery or 3-5 weeks after mastectomy, patients undergo radiotherapy daily 5 days a week for 6-8 weeks. Patients receive long-term trastuzumab IV over 30-90 minutes weekly for 40 weeks beginning on week 36 (day 254).